Random fusion frames are nearly equiangular and tight

TL;DR

This paper shows that randomly chosen subspaces in a real Hilbert space form nearly tight and nearly equiangular fusion frames, with high probability, under certain dimension constraints, using measure concentration arguments.

Contribution

It establishes that random, unitarily invariant subspaces are nearly tight and equiangular fusion frames, providing probabilistic guarantees based on dimension ratios.

Findings

Random subspaces form nearly tight fusion frames.

Subspaces are nearly equiangular with high probability.

Dimension constraints ensure success probability.

Abstract

This paper demonstrates that random, independently chosen equi-dimensional subspaces with a unitarily invariant distribution in a real Hilbert space provide nearly tight, nearly equiangular fusion frames. The angle between a pair of subspaces is measured in terms of the Hilbert-Schmidt inner product of the corresponding orthogonal projections. If the subspaces are selected at random, then a measure concentration argument shows that these inner products concentrate near an average value. Overwhelming success probability for near tightness and equiangularity is guaranteed if the dimension of the subspaces is sufficiently small compared to that of the Hilbert space and if the dimension of the Hilbert space is small compared to the sum of all subspace dimensions.